Event Description:Thomas Garrity, William R. Kenan Jr. Professor of Mathematics at Williams College, will give an invited colloquium mathematics lecture as part of the Department of Mathematical Sciences' annual "Celebration of the Mind."

Abstract: Most of us use number almost everyday. But, overall, numbers and their properties are still quite mysterious. In fact, to some extent, it is not even known what is the best way to way to write down a number. Can we write numbers so that by just looking at them, we can tell that they are special? For example, most of us usually express real numbers in terms of their decimal expansions. Here, a real number will be rational precisely when its decimal expansion is eventually periodic.

There are other ways of expressing numbers, though. For example, one can also write a real number in terms of its continued fraction expansion, which we will be defining. A number's continued fraction expansion will be eventually periodic precisely when the number is a quadratic irrational (i.e., a number involving a square root).

But what about other types of numbers, such as cube roots, fourth roots and other algebraic numbers? Is there some way of expressing real numbers as a sequence of integers such that periodicity is equivalent to being a cubic irrational or some other type of algebraic number? This is the Hermite Problem, which, in 1848, Hermite posed to Jacobi. While there have been many attempts to develop algorithms to solve this problem over the years, it is still open, though that has been true progress in the last few years.

The only real prerequisite for this talk is to know a bit about fractions.